Progeria (and related) syndromes are essentially a collection of 'accelerated aging' phenotypes caused by single mutations; Progerin is a shortened version of the protein Lamin A, and is therefore not found in individuals without a loss-of-function mutation in the LMNA gene (the wiki page you reference). As far as we are aware these genes do not 'cause' aging in individuals without the mutations.
LMNA is a normal component of the nuclear lamina (a structure inherent to the nucleus). This review discusses the various diseases associated with mutations in this gene, some of which present 'accelerated aging' phenotypes. However, as far as I know, there is limited evidence to suggest that LMNA, or indeed any Lamina associated protein, is involved in 'normal aging'. A recent GWAS meta-analysis found a variant in LMNA that is associated with longevity in humans, however the association is relatively weak (OR=1.18, P=7(x10)-4), so even if this is a true association, it seems that (as usual in aging research) there are many other factors to consider, and it is not a single gene that is doing the aging.
So to stress the point: progerin has no function in 'normal' human aging - it is a defective protein caused by a germline (or novel) mutation in the LMNA gene. Accelerated aging is the symptom of this genetic disorder, and is not completely analogous to normal aging (as you point out, there is no cognitive decline that is associated with normal human aging).